The Leavitt Path Algebras of Generalized Cayley Graphs
Let n be a positive integer. For each 0 ≤ j ≤ n - 1 , we let C n j denote Cayley graph for the cyclic group Z n with respect to the subset { 1 , j } . For any such pair ( n , j ), we compute the size of the Grothendieck group of the Leavitt path algebra L K ( C n j ) ; the analysis is related to a c...
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Published in | Mediterranean journal of mathematics Vol. 13; no. 1; pp. 1 - 27 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Cham
Springer International Publishing
01.02.2016
|
Subjects | |
Online Access | Get full text |
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Summary: | Let
n
be a positive integer. For each
0
≤
j
≤
n
-
1
, we let
C
n
j
denote Cayley graph for the cyclic group
Z
n
with respect to the subset
{
1
,
j
}
. For any such pair (
n
,
j
), we compute the size of the Grothendieck group of the Leavitt path algebra
L
K
(
C
n
j
)
; the analysis is related to a collection of integer sequences described by Haselgrove in the 1940s. When
j
= 0, 1, or 2, we are able to extract enough additional information about the structure of these Grothendieck groups so that we may apply a Kirchberg-Phillips-type result to explicitly realize the algebras
L
K
(
C
n
j
)
as the Leavitt path algebras of graphs having at most three vertices. The analysis in the
j
= 2 case leads us to some perhaps surprising and apparently nontrivial connections to the classical Fibonacci sequence. |
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ISSN: | 1660-5446 1660-5454 |
DOI: | 10.1007/s00009-014-0464-4 |